Question

A young couple wants to have a college fund that will pay $30,000 at the end of each half-year for 8 years. (a) If they can invest at 7%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years to begin making their college withdrawals 6 months after their last investment? (Round your answer to the nearest cent.) (b) Suppose 8 years after beginning the annuity payments, they receive an inheritance of $36,000 that they contribute to the account, and they continue to make their regular payments as found in part (a). How many college withdrawals will they be able to make before the account balance is $0? (Round your answer to the nearest whole number.) withdrawals Need Help? Read It Talk to a Tutor

Answer 1

Answers:
a. $23,064.08
b. 21 withdrawals.

a. Formula: The present value of an ordinary annuity (PV)

PV = C× [1-(1+r)^-n]/r

PV = Present value (The cummulative amount available at starting of college)
C= Periodic cash flow.
r =effective interest rate for the period.
n = number of periods.

PV = 30,000× [1-(1+0.035)^-16]/0.035
PV = 362,823.50

C= Periodic cash inflow. $30000
r =effective interest rate for the period. 7%÷2 = 3.5%= 0.035
n = number of periods. 8x2= 16

Formula: The Future Value of an ordinary annuity (FV)

FV= C× {[(1+r)^n]-1}/r

FV = Future value (The cumulative amount available at starting of college)
C= Periodic cash outflow.
r =effective interest rate for the period.
n = number of periods.

Formula: The Future Value of an ordinary annuity (FV)

362,823.50= C× {[(1+0.035)^36]-1}/0.035

C= $23,064.08

FV = $362,823.50
C= Periodic cash out flow needed=?
r =effective interest rate for the period. 7÷2= 3.75%
n = number of periods. 18×2= 36

Answer:
How much should she deposit each 6 months? = $23,064.08

b.
Future value 36,000 fourtune received:

Future value= present value (1+r)^n
FV = 36,000(1.035)^20
FV = 71,632.39

Total amount available at the starting of the college= 71,632.39+362,823.50 =434,455.89

434455.89 = 30,000× [1-(1+0.035)^-n]/0.035

PV = 434,455.89 (The cummulative amount available at starting of college)
C= Periodic cash flow. 30,000
r =effective interest rate for the period. 7÷2= 3.5% =0.035
n = number of periods.

N= 20.55
N= 21 periods

Answer: they can have 21 withdrawals.